Garibaldi temperament

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Garibaldi temperament is a 7-limit (and higher) temperament of the schismatic family. It is an extension of helmholtz temperament beyond the 5-limit but with the same simple chain-of-fifths structure (so that standard notation may be used). As in helmholtz temperament, 5/4 is mapped to the diminished fourth (e.g. A-D♭), and the new mapping specific to garibaldi is that 7/4 is mapped to the double diminished octave (e.g. A-Abb). This makes garibaldi a marvel temperament.

Immediate 11-limit extensions include cassandra (41&53), mapping 11/8 to +23 fifths, andromeda (29&41), mapping 11/8 to -18 fifths, and helenus (53&65d), mapping 11/8 to -30 fifths.

Interval chain

In the following table, prime harmonics are in bold.

Fifth
generator 		Cents* Approximate Ratios
7-limit 13-limit Extension
Cassandra Andromeda Helenus
0 0.00 1/1
1 702.09 3/2
2 204.17 9/8
3 906.26 27/16, 42/25 22/13 22/13 22/13
4 408.34 63/50, 80/63 14/11
5 1110.43 40/21 21/11
6 612.51 10/7
7 114.60 15/14, 16/15 14/13
8 816.68 8/5 21/13
9 318.77 6/5 40/33
10 1020.85 9/5 20/11
11 522.94 27/20 15/11
12 25.02 50/49, 64/63, 81/80 40/39, 45/44
13 727.11 32/21 20/13
14 229.19 8/7 15/13
15 931.28 12/7
16 433.36 9/7 14/11
17 1135.45 27/14, 48/25 52/27 64/33 21/11
18 637.53 36/25, 81/56 13/9 16/11
19 139.62 27/25 13/12 12/11 14/13
20 841.70 80/49, 81/50 13/8, 44/27 18/11, 64/39 21/13
21 343.79 60/49 11/9, 39/32 16/13, 27/22 40/33
22 1045.87 64/35 11/6 24/13 20/11
23 547.96 48/35 11/8 18/13 15/11
24 50.04 36/35 33/32 27/26 40/39, 45/44
25 752.13 54/35 20/13
26 254.21 81/70, 144/125 15/13
27 956.30 216/125, 256/147 26/15
28 458.38 64/49 13/10
29 1160.47 96/49 39/20, 88/45 64/33
30 662.55 16/11
31 164.64 12/11
32 866.72 18/11, 64/39
33 368.81 16/13, 27/22
34 1070.90 24/13
35 572.98 18/13

* in 7-limit POTE tuning

Tuning spectra

Cassandra

Gencom: [2 4/3; 225/224 275/273 325/324 385/384]

Gencom mapping: [⟨1 2 -1 -3 13 12], ⟨0 -1 8 14 -23 -20]]

Eigenmonzo
(Unchanged-interval) 		Generator
(¢) 		Comments
16/15 701.676
5/4 701.711
[0 -10 17⟩ 701.728 5-odd-limit least squares
6/5 701.738 5-odd-limit minimax
10/9 701.760
15/13 701.9355
13/10 701.9362
4/3 701.955
16/13 702.026
13/12 702.030
18/13 702.034
11/10 702.097
15/11 702.102
14/13 702.109 13 and 15-odd-limit minimax
[0 -95 -137 -129 167 143⟩ 702.112 15-odd-limit least squares
[0 -27 7 17⟩ 702.114 9-odd-limit least squares
[0 -38 -80 -122 137 116⟩ 702.128 13-odd-limit least squares
[0 -25 11 35⟩ 702.140 7-odd-limit least squares
[0 17 -52 -88 134⟩ 702.183 11-odd-limit least squares
9/7 702.193 9 and 11-odd-limit minimax
7/6 702.209 7-odd-limit minimax
8/7 702.227
14/11 702.230
11/8 702.231
12/11 702.244
11/9 702.258
15/14 702.778
7/5 702.915
13/11 703.597

Andromeda

Gencom: [2 4/3; 100/99 105/104 196/195 245/242]

Gencom mapping: [⟨1 2 -1 -3 -4 -5], ⟨0 -1 8 14 18 21]]

Eigenmonzo
(Unchanged-interval) 		Generator
(¢) 		Comments
16/15 701.676
5/4 701.711
6/5 701.738 5-odd-limit minimax
10/9 701.760
4/3 701.955
9/7 702.193 9-odd-limit minimax
7/6 702.209 7-odd-limit minimax
8/7 702.227
11/9 702.630 11-odd-limit minimax
12/11 702.665
11/8 702.705
18/13 702.756 13 and 15-odd-limit minimax
15/14 702.778
13/12 702.792
16/13 702.832
7/5 702.915
15/11 703.359
15/13 703.410
11/10 703.500
13/10 703.522
13/11 703.597
14/13 704.043
14/11 704.377

Helenus

Gencom: [2 4/3; 99/98 176/175 275/273 847/845]

Gencom mapping: [⟨1 2 -1 -3 -9 -10], ⟨0 -1 8 14 30 33]]

Eigenmonzo
(Unchanged-interval) 		Fifth
(¢) 		Comments
14/11 701.094
14/13 701.489
11/10 701.591
15/11 701.607
11/8 701.623
12/11 701.633
11/9 701.644 11, 13, and 15-odd-limit minimax
16/15 701.676
5/4 701.711
6/5 701.738 5-odd-limit minimax
10/9 701.760
16/13 701.802
13/12 701.807
18/13 701.811
13/10 701.831
15/13 701.836
4/3 701.955
9/7 702.193 9-odd-limit minimax
7/6 702.209 7-odd-limit minimax
8/7 702.227
15/14 702.778
7/5 702.915
13/11 703.597

Scales

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